SOCIAL WELFARE FUNCTION

PROF. PRABHA PANTH,OSMANIA UNIVERSITY,

HYDERABAD

Bergson and Samuelson’s Social Welfare FunctionHicks and Kaldor’s Compensation principle – does

not show a unique equilibrium point, where individuals’ and social welfare is maximised.

If Ps of the two goods change, then equilibrium point also changes.

Does the new equilibrium give greater welfare than the old?

Also they have indirectly brought in Cardinal Utility, which is not measureable.

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Each economic unit wants to maximise its welfare.Consumers’ welfare = maximise utility,Producers’ welfare = maximise profits,Factors’ welfare = maximise incomes.

Is the sum of maximum individual welfare = maximum social welfare?

i.e. Ui = SW (i = 1,2,….. N)Welfare Economics is based on Normative Economics, Equity Principles, Value Judgments

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Pareto OptimalityPareto Optimality: Each individual economic unit

maximises its welfare.But if X’s welfare increases, and Y’s welfare

decreases, what happens to Social Welfare (SW)? Does it increase or decrease?In the Pareto system there is no unique equilibrium.If relative prices change, leads to new equilibrium.Does this new equilibrium denote improvement in

SW? See Figure 1.

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5

Com

mod

ity B

P1A/

P1B

F

Commodity A

0

E

PA/PB

•Original equilibrium is at E, on the blue PPC.•Price ratio is PA/PB.• Due to change in economic conditions, PPC may change to the Red curve.•Price ratios have also changed to P1A/P1B.•New equilibrium is at F.•What has happened to efficiency?•Equity?•Social welfare at F compared to E?•Is there any improvement? •PPC shows only production efficiency, not of consumption.

FIGURE 1

Social Welfare Function - SWFBergson: SWF is an ordinal index of society’s welfare.It is a function of the utility levels of all individuals in

society. Brings in optimisation of consumption welfare also.

Pareto optimum reflected only production optimisation.

Based on Value Judgment of each individual,Reflected in consumption,Choices are transitive and consistent.SWF = W(U1, U2, ............... UN)

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The Social Indifference Curve

7

N’s

util

ity

inde

x

M’s utility index

0

W1W2

W3

Q

T

R

S

The Social Indifference Curve: is a locus of various combinations of UM and UN, which results in equal level of Social Welfare.In the figure, W1, W2, W3, are Social Indifference Curves.Each curve shows equal different combinations of utility of the two consumers M and N, which give the same level of Social Welfare. The Ws have same properties as

ordinary Indifference Curves.All combinations on each W, have same level of Social Welfare.Higher W has higher level of Social Welfare.

From R on W1 to T on W2 N’s U, M’s . But on higher W, so Social Welfare increases.Same case of movement from T to Q on W3. M’s U and N’s U. But Social Welfare as shift to higher W.From T to S, SW constant, as they are on same W.

FIGURE 2

Utility Possibility Curve In Fig.3, the contract curve of consumers 0A 0B, shows equilibrium points of the two consumers. Moving from point 1 to 4, as UX , UY Plotting the contract curve on a diagram, gives the Utility Possibility Curve or Frontier for a given set of Ps. Figure 4. If Ps change, so will the Utility Possibility curve,

8

IC 1X

IC2 X

IC3 X

IC4X

BY

IC2Y

IC3Y

IC4Y

AX

IC1Y

1

23

4

AY

BX

0B

0A

UX

3UX

1 UX

2

UY

4O

UY

UY1UY2

UY3

UY4

1

2

3

4

Figure 3 Figure 4Utility Possibility Frontier

Grand Utility Possibility Frontier

9

O

UY

F1G1H1

F1

UX

G1 H1

Figure 5The Grand Utility Possibility Frontier

• When Ps change, the contract curve changes, and the Utility Possibility Curve also changes.(F1F1, G1G1, and H1H1). • The Outer Envelope of the different Utility Possibility Frontier curves is called “The Grand Utility Possibility Frontier.”• It is the locus of all the possible contract curves or Utility Possible curves from changes in the prices of the two goods.• Satisfies the marginal conditions: MRTS = MRS for each commodity mix.• Satisfies Pareto marginal conditions of a) efficiency in production, b) efficiency in distribution, and c) efficiency in product composition.

Point of Constrained BlissBut again this does not give a Unique Point of

Social Welfare.If the SWF and Grand Utility Possibility Frontier are

combined, then the point of maximum Social and Individual together can be found’

This point is called the Point of Constrained Bliss.Because a movement away from it will reduce

Social Welfare.

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O

UY

W*

G

H

U*Y

UX

U*X

H1

Constrained Bliss Point

W1

W2W3

W4

Figure 6

In Fig.6 the Grand Utility Frontier is combined with a set of social indifference curves (Ws). Social welfare is maximised at W*, where Grand Utility Possibility Frontier is a tangent to the highest possible Social Indifference curve W3. The two consumers enjoy the highest levels of welfare of U*Y and U*X.Point G on W1 is on the Grand Utility Frontier, but not unique equilibrium point.Similarly H on a higher Social Welfare curve than G, but not an equilibrium point.W* is the maximum social welfare possible, given factor endowments, state of technology, and individual’s preferences. At W*, Social Justice condition: Slope of Iso welfare curve = Slope of Utility Frontier